Conditioning by equal, linear
Author:
Chii-Ruey Hwang
Journal:
Trans. Amer. Math. Soc. 274 (1982), 69-83
MSC:
Primary 60B99; Secondary 68G10
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670919-9
MathSciNet review:
670919
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Abstract | References | Similar Articles | Additional Information
Abstract: We deal with a limit problem of regularity controlled probabilities in metric pattern theory. The probability on the generator space is given by a density function on which some integrability conditions are imposed. Let
denote the integral operator with kernel
. When
i.i.d. generators
are connected together to form the configuration space
via the regularity
, i.e. "conditioning" on
for
, an approximate identity is used to define the regularity controlled probability on
. The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration
is described by its corresponding marginal probability within
. When
goes to infinity in a suitable way, the above mentioned marginal probability converges weakly to a limit whose density can be expressed in terms of the largest eigenvalues and the corresponding eigenspaces of
and
. When
is bivariate normal, the eigenvalue problem is solved explicitly. The process determined by the limiting marginal probabilities is strictly stationary and Markovian.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670919-9
Keywords:
Approximate identity,
bivariate Gaussian,
compact positive operator,
conditioning,
configuration,
eigenvalue,
eigenfunction,
generator,
Hermite polynomial,
integral equation,
Markovian,
normal operator,
pattern theory,
regularity controlled probability,
strictly stationary,
weak convergence
Article copyright:
© Copyright 1982
American Mathematical Society